Tikalon Weblog by Dev Gualtieri

Koch Snowflake

November 6, 2023

You Bet Your Life was a well-liked game show hosted by Groucho Marx (1890-1977), first on radio, then on television, in the course of the 1940s and 1950s. To make sure that each contestant would go away with some prize, Marx would ask a query resembling, “Who’s buried in Grant’s tomb?” Grant’s tomb is a big mausoleum in New York City containing the stays of American Civil War General, Ulysses S. Grant (1822-1885), and his wife. The accepted reply was Grant. Nonetheless, Grant and his spouse will not be truly beneath ground within the mausoleum; so, nobody is definitely buried in Grant’s tomb.[1]

I used to be reminded of this query after I learn a latest arXiv paper by Yann Demichel of the Université Paris Nanterre entitled, Who invented von Koch’s Snowflake Curve?.[2] As Demichel writes, “…the image of the snowflake curve shouldn’t be current and even talked about in von Koch’s unique articles.”[2] Whereas the 1904 paper, “Sur une courbe proceed sans tangente, obtenue par une development géométrique élémentaire” (On a steady curve with out tangents constructible from elementary geometry), cited as its origin has a number of figures, none of those are the snowflake.[3]

The mathematical object referred to as the Koch snowflake is a fractal curve credited to Swedish mathematician, Helge von Koch (1870-1924). The curve has a fractal dimension (Hausdorff dimension) of log 4/log 3, about 1.26. The snowflake, as proven within the determine, is just constructed, as follows:[4]

• Draw an equilateral triangle.
• Taking every line segment in flip, divide it into three equal segments.
• Draw an equilateral triangle pointing outwards utilizing the center section because the base.
• Take away the bottom.
• Proceed this operation for the subsequent two line segments of the unique triangle; after which recursively for all line segments.

The Koch snowflake in its first-fourth iteration. The beginning determine, an equilateral triangle, is omitted.

This fractal is far simpler to generate and visualize than the favored Mandelbrot set, which has a Hausdorff dimension of two.

(Rendered utilizing Inkscape from this Python program. Click for larger image.)

As for who was the primary to provide the Koch snowflake, Demichel traces its first instance to a 1940 popular science book, Arithmetic and Creativeness, co-authored by Edward Kasner (1878-1955), a professor at Columbia University (New York, NY), and one in all his students.[2] The snowflake is proven in an appendix entitled, Pathological Curves, however von Koch shouldn’t be talked about.[2] Demichel additional speculates that the snowflake identify may need been prompt by a child in Kasner’s family, and that Kasner may have designed the snowflake curve previous to 1916.[2]

A well-liked hobby amongst amateur radio operators is making radio antennas from uncommon objects. The primary fractal antenna was patented in 2002,[5] and there is an online exposition of a Koch snowflake antenna with fascinating broadband properties[6] (see determine).

Left, determine 7E of US Patent No. 6,452,553, “Fractal antennas and fractal resonators,” by Nathan Cohen, September 17, 2002; and, proper, a Koch snowflake antenna. Engineers will discover the resemblance of the antenna on the left to a twin-lead “T” antenna. (Left picture by way of Google Patents,[5] and proper picture (modified) from Ref. 6.[6] Click for larger image.)

I wrote about some fascinating properties of precise snowflakes in an older article (Snowflakes, January 18, 2011). The foremost query about snowflakes is whether or not it is actually true that no two snowflakes are alike. On the atomic degree, a snowflake will include about 10¹⁸ water molecules, a few of which is perhaps shaped from deuterium, relatively than hydrogen. On Earth, there’s one deuterium atom for each 6,420 atoms of hydrogen in water. There is no must calculate the permutation, because it’s intuitive that even with the estimated 10²⁴ snowflakes falling per year, the odds of any two matching is indistinguishable from zero.[7]

Hooke’s 1665 e book, Micrographia, included many drawings of snowflakes, and these have been the primary drawings to point the complex shapes inside shapes in snowflakes. Greater than 200 years later, beginning in 1885, after the invention of photography, Vermont farmer, Wilson Alwyn Bentley, took quite a few photomicrographs of snowflakes. His images documented the high-quality particulars inherent in snowflakes and confirmed anecdotally how unlikely that two might be the identical (see determine).

See Also

On Internet-Safety and -Insecurity: Insecure Options in PDFs

Snowflakes photographed at Jericho, Vermont, by Wilson Alwyn Bentley, 1902. (Wikimedia Commons images.)

References:

1. Quiz Question: Who Is Buried in Grant’s Tomb? Answer: Grant!, Quote Investigator, November 10, 2011.
2. Yann Demichel, “Who invented von Koch’s Snowflake Curve?” arXiv, August 29, 2023, https://doi.org/10.48550/arXiv.2308.15093.
3. Helge von Koch, “Sur une courbe continue sans tangente, obtenue par une construction géométrique élémentaire”. Ark. Mat. Astron. Fys., vol. 1 (1904), pp. 681-704.
4. Koch Snowflake page on Wikipedia.
5. Nathan Cohen, “Fractal antennas and fractal resonators,” US Patent No. 6,452,553 , September 17, 2002.
6. KB7QHC, “Koch Snowflake Antenna,” QSL.net.
7. SnowCrystals.com, created by Kenneth G. Libbrecht of Caltech.

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